If we consider a satellite in orbit around the earth, the satellite has an infinite number of different potential energies: one for each possible frame of reference we could choose.
Correct. And the kinetic energy is also different for different frames. A conserved quantity is one that is the same at two moments. An invariant quantity is something that is the same for all observers. They are different concepts. Energy is a conserved quantity, not an invariant quantity.
Same with momentum.
If we choose one reference frame first, and then shift the reference frame to another one, then we also instantly change the potential energy of the object.
No, the object never had potential energy, the system did. But sure, things can change when you change frames. Just like the center of mass could be at the origin and then if you pick a new coordinate system the center of mass might no longer be at the origin. The same thing is going on.
And no physics is going on. You can just measuring relative to a different standard.
However, this change doesn't correspond with any change in reality.
Exactly, things that are not frame invariant, such as energy or being at the origin, can be different in different frames. And this doesn't correspond to reality. So don't get too excited about things that depend on tour frame.
That said you can still say things like these particles don't have enough relative velocity to separate beyond this distance because of the rise in potential energy the system gets as they move apart.
One possible defence could be to say that an object has all its possible potential energies at the same time.
Again, the particle doesn't have potential energy, the system does. In introductory classes where they are looking at the simplest possible systems and only one thing is moving then the location of the thing tells you the state of the system. But the potential energy is a property of the system.
However, if we look at all possible potential energies of an object (say, by summing all the potential energies between the object and all other objects on the universe),
You talked about different energies in different frames noe you want to talk about interactions between different objects, maybe you are confused about even more issues.
we get a measure that is constantly increasing (the universe is expanding)
This makes no sense. Some interactions are attractive and others are repulsive, so an expanding distance between objects could have kinetic energy go up or go down.
and doesn't change in a locally appropriate way
Potential energy is not stored in particles, it is a property of the system so it is non local.
having the satellite decrease its height above earth should decrease its potential energy.
The earth-satellite system has less potential potential energy when they are closer together.
However, if all other possible potential energies are factored in, that's probably not what happens).
There is no probably about it: the potential energy of the system of all the masses is $U=-\sum_{i\neq j}Gm_im_j/r_{ij}$ where $r_{ij}$ is the distance between $m_i$ and $m_j.$ Whether the energy gets lower depends on whether you two closer together more than you made them get farther away from others.
This would suggest that the energy of the universe is constantly increasing, as distances in the universe are constantly increasing.
I can't see how you can conclude anything like that. Energy is conserved the fact that it is frame dependent is irrelevant to the fact that it doesn't change its total over time.
And if you want to take the expanding universe into account you need general relativity, which doesn't use potential energy becsuse it needs a local version of energy and then energy is not conserved and there isn't a total energy anyway since energy depends on frame and there is no global frame.
I don't see how one could meaningfully say that potential energy corresponds with reality.
Well, sometimes even in general relativity you find a constant of the motion (something that can be computed in a coordinate system that doesn't change) where is looks just like kinetic plus potential energy, so it does come up even when you weren't looking for it. And then people make things called Lagrangians that have units of energy that act similar to potential energies in the sense that it generates the dynamics and kinematics and motion, these can also lead to Hamiltonians that can be conserved over time and have units of energy. And finally. The thibgs that people represent with potential energy can sometimes be replaced with having energy stored in fields (so then the energy is conserved and it goes someplace, moves around, and go go back to types of objects that gave the fields the energy).
So thibgs like it come up and there are ways it generalizes that is still useful. So its a bit harsh. And summing over different frames isn't a thing and invariant is different than conserved.
